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%I #39 Sep 09 2023 19:09:55
%S 1,1,5,29,182,1198,8142,56620,400598,2872754,20824778,152303410,
%T 1122149800,8319825040,62017475600,464452683432,3492568119566,
%U 26358270711370,199565061455634,1515311001158482,11535716330003876,88025068713285476,673124069796140900
%N Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.
%H Seiichi Manyama, <a href="/A190917/b190917.txt">Table of n, a(n) for n = 0..1111</a> (terms 1..200 from R. H. Hardin)
%H R. J. Mathar, <a href="http://vixra.org/abs/1511.0015">A class of multinomial permutations avoiding object clusters</a>, vixra:1511.0015 (2015), sequence M_{3,m}/3!.
%F a(n) = A110706(n) / 6 for n >= 1.
%F n*(n+1)*a(n) - (n+1)*(7*n-4)*a(n-1) - 8*(n-2)^2*a(n-2) = 0. - _R. J. Mathar_, Nov 01 2015 from A110706
%e All solutions for n=2:
%e 1 1 1 1 1
%e 2 2 2 2 2
%e 3 3 3 3 1
%e 1 2 2 1 3
%e 3 3 1 2 2
%e 2 1 3 3 3
%p a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1],
%p ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
%p end:
%p seq(a(n), n=0..22); # _Alois P. Heinz_, Sep 09 2023
%t Table[(1/3)*Sum[Binomial[n-1, k]*(Binomial[n-1, k]*Binomial[2*n+1-2*k, n+1] + Binomial[n-1, k+1]*Binomial[2*n-2*k, n+1]), {k,0,Floor[n/2]}], {n,1,25}] (* _G. C. Greubel_, Nov 24 2018 *)
%o (PARI) A190917(n) = sum(k=0, n\2, binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1))) / 3; \\ _Max Alekseyev_, Dec 10 2017
%o (Magma) [(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]])/3: n in [1..25]]; // _G. C. Greubel_, Nov 24 2018
%o (Sage) [(1/3)*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1)) for k in range(1+floor(n/2))) for n in (1..25)] # _G. C. Greubel_, Nov 24 2018
%Y Column 3 of A322013.
%Y Cf. A000012 (b=2), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9), A322061 (b=10).
%K nonn
%O 0,3
%A _R. H. Hardin_, May 23 2011
%E a(0)=1 prepended by _Alois P. Heinz_, Sep 09 2023
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